Question

Evaluate \(f(x)=\frac{5}{9} x^2-\frac{2}{3} x-7\) for \(x=-6\) A. 7 B. 13 C. 17 D. 20

Short answer

The correct answer is C. 17

Step-by-step solution

Substitute the Value

Substitute the given value of x into the function. Here, we need to evaluate the function for \(x = -6\). So, replace \(x\) with \(-6\) in the expression:\[ f(-6) = \frac{5}{9}(-6)^2 - \frac{2}{3}(-6) - 7 \]

Simplify the Squared Term

First, calculate \((-6)^2\):\[ (-6)^2 = 36 \]So the expression becomes:\[ f(-6) = \frac{5}{9} \times 36 - \frac{2}{3}(-6) - 7 \]

Simplify the First Fraction

Now, simplify \(\frac{5}{9} \times 36\):\[ \frac{5}{9} \times 36 = 5 \times 4 = 20 \]

Simplify the Second Fraction

Next, simplify \( -\frac{2}{3}(-6) \):\[ -\frac{2}{3} \times (-6) = 2 \times 2 = 4 \]

Combine the Results

Now combine all the simplified values into the expression:\[ f(-6) = 20 + 4 - 7 \]

Final Calculation

Finally, perform the arithmetic operations to find:\[ 20 + 4 - 7 = 17 \]

Key concepts

quadratic functions

A quadratic function is a special type of polynomial function. It’s written in the form: \[ f(x) = ax^2 + bx + c \]where 'a', 'b', and 'c' are constants and 'x' represents an unknown variable. The value of 'a' cannot be zero because if 'a' is zero, the function would become linear, not quadratic.
One of the key features of a quadratic function is its parabolic shape when graphed. This means it either opens upwards (when 'a' is positive) or downwards (when 'a' is negative). Quadratic functions often appear in various fields, including physics for calculating projectile motions and in finance for modeling certain behaviors.

substitution method

The substitution method is a simple, yet powerful technique used in mathematics to solve equations. In this method, you replace a variable with a given value to find the solution.
For example, in the exercise, we are given the quadratic function\( f(x) = \frac{5}{9}x^2 - \frac{2}{3}x - 7 \)and we need to evaluate it for \( x = -6 \). This means substituting every instance of ‘x’ in the equation with '-6'. Here's how that works:

• Replace ‘x’ in the function expression with '-6':\[ f(-6) = \frac{5}{9}(-6)^2 - \frac{2}{3}(-6) - 7 \]
• By doing this, we transform the function into a numerical expression that can be simplified step-by-step.

simplification

Simplification refers to the process of reducing a mathematical expression to its simplest form. Let's follow the steps from our given solution to understand this concept:

• Start by substituting '-6' into the function:\[ f(-6) = \frac{5}{9}(-6)^2 - \frac{2}{3}(-6) - 7 \]
• Next, simplify the squared term: \[ (-6)^2 = 36 \]So the expression becomes:\[ f(-6) = \frac{5}{9} \times 36 - \frac{2}{3}(-6) - 7 \]
• We then simplify the fraction \( \frac{5}{9} \times 36 \):\[ \frac{5}{9} \times 36 = 5 \times 4 = 20 \]
• Next, simplify \( -\frac{2}{3}(-6) \):\[ -\frac{2}{3} \times (-6) = 2 \times 2 = 4 \]
• Now, combine all the simplified values into the expression:\[ f(-6) = 20 + 4 - 7 \]
• Finally, perform the arithmetic to reach the final answer:\[ 20 + 4 - 7 = 17 \]

Notice how breaking down the problem into smaller steps makes the process clearer and easier to understand. Simplification helps in reducing complexity and making calculations more manageable.